Method of determining resistivity of an earth formation with phase resistivity evaluation based on a phase shift measurement and attenuation resistivity evaluation based on an attenuation measurement and the phase shift measurement

ABSTRACT

A resistivity measurement technique estimates a first value for a first electrical parameter consistent with an assumption that each property of a measured electrical signal senses the first electrical parameter and a second electrical parameter in substantially the same volume and estimates a second value of the first electrical parameter consistent with the estimated first value and consistent with each property of the measured electrical signal sensing the first electrical parameter and the second electrical parameter in different volumes. Applying this technique, a phase conductivity may be determined from only a phase shift measurement. An attenuation conductivity may then be determined based on an attenuation measurement and the phase conductivity. Since bandwidth is limited in data telemetry to an earth surface while drilling, phase shift measurements can be telemetered without attenuation measurements for accomplishing resistivity measurements.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. patent application,Ser. No. 09/608,205, filed Jun. 30, 2000, U.S. Pat. No. 6,366,858,entitled “Method of and Apparatus for Independently Determining theResistivity and/or Dielectric Constant of an Earth Formation,” to S.Mark Haugland, which is hereby incorporated by reference in its entiretyfor all purposes.

STATEMENTS REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Not applicable.

REFERENCE TO A MICROFICHE APPENDIX

Not applicable.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention generally relates to a method of surveying earthformations in a borehole and, more specifically, to a method ofdetermining the resistivity of earth formations with phase resistivityevaluation based on phase shift measurement and attenuation resistivityevaluation based on an attenuation measurement and the phase shiftmeasurement in connection with Measurement-WhileDrilling/Logging-While-Drilling and Wireline Logging operations.

2. Description of the Related Art

Typical petroleum drilling operations employ a number of techniques togather information about earth formations during and in conjunction withdrilling operations such as Wireline Logging, Measurement-While-Drilling(MWD) and Logging-While-Drilling (LWD) operations. Physical values suchas the electrical conductivity and the dielectric constant of an earthformation can indicate either the presence or absence of oil-bearingstructures near a drill hole, or “borehole.” A wealth of otherinformation that is useful for oil well drilling and production isfrequently derived from such measurements. Originally, a drill pipe anda drill bit were pulled from the borehole and then instruments wereinserted into the hole in order to collect information about down holeconditions. This technique, or “wireline logging,” can be expensive interms of both money and time. In addition, wireline data may be of poorquality and difficult to interpret due to deterioration of the regionnear the borehole after drilling. These factors lead to the developmentof Logging-While-Drilling (LWD). LWD operations involve collecting thesame type of information as wireline logging without the need to pullthe drilling apparatus from the borehole. Since the data are taken whiledrilling, the measurements are often more representative of virginformation conditions because the near-borehole region often deterioratesover time after the well is drilled. For example, the drilling fluidoften penetrates or invades the rock over time, making it more difficultto determine whether the fluids observed within the rock are naturallyoccurring or drilling induced. Data acquired while drilling are oftenused to aid the drilling process. For example, MWD/LWD data can help adriller navigate the well so that the borehole is ideally positionedwithin an oil bearing structure. The distinction between LWD and MWD isnot always obvious, but MWD usually refers to measurements taken for thepurpose of drilling the well (such as navigation) whereas LWD isprincipally for the purpose of estimating the fluid production from theearth formation. These terms will hereafter be used synonymously andreferred to collectively as “MWD/LWD.”

In wireline logging, wireline induction measurements are commonly usedto gather information used to calculate the electrical conductivity, orits inverse resistivity. See for example U.S. Pat. No. 5,157,605. Adielectric wireline tool is used to determine the dielectric constantand/or resistivity of an earth formation. This is typically done usingmeasurements which are sensitive to the volume near the borehole wall.See for example U.S. Pat. No. 3,944,910. In MWD/LWD, a MWD/LWDresistivity tool is typically employed. Such devices are often called“propagation resistivity” or “wave resistivity” tools, and they operateat frequencies high enough that the measurement is sensitive to thedielectric constant under conditions of either high resistivity or alarge dielectric constant. See for example U.S. Pat. Nos. 4,899,112 and4,968,940. In MWD applications, resistivity measurements may be used forthe purpose of evaluating the position of the borehole with respect toboundaries of the reservoir such as with respect to a nearby shale bed.The same resistivity tools used for LWD may also used for MWD; but, inLWD, other formation evaluation measurements including density andporosity are typically employed.

For purposes of this disclosure, the terms “resistivity” and“conductivity” will be used interchangeably with the understanding thatthey are inverses of each other and the measurement of either can beconverted into the other by means of simple mathematical calculations.The terms “depth,” “point(s) along the borehole,” and “distance alongthe borehole axis” will also be used interchangeably. Since the boreholeaxis may be tilted with respect to the vertical, it is sometimesnecessary to distinguish between the vertical depth and distance alongthe borehole axis. Should the vertical depth be referred to, it will beexplicitly referred to as the “vertical depth.”

Typically, the electrical conductivity of an earth formation is notmeasured directly. It is instead inferred from other measurements eithertaken during (MWD/LWD) or after (Wireline Logging) the drillingoperation. In typical embodiments of MWD/LWD resistivity devices, thedirect measurements are the magnitude and the phase shift of atransmitted electrical signal traveling past a receiver array. See forexample U.S. Pat. Nos. 4,899,112, 4,968,940, or 5,811,973. In commonlypracticed embodiments, the transmitter emits electrical signals offrequencies typically between four hundred thousand and two millioncycles per second (0.4-2.0 MHz). Two induction coils spaced along theaxis of the drill collar having magnetic moments substantially parallelto the axis of the drill collar typically comprise the receiver array.The transmitter is typically an induction coil spaced along the axis ofa drill collar from the receiver with its magnetic moment substantiallyparallel to the axis of the drill collar. A frequently used mode ofoperation is to energize the transmitter for a long enough time toresult in the signal being essentially a continuous wave (only afraction of a second is needed at typical frequencies of operation). Themagnitude and phase of the signal at one receiving coil is recordedrelative to its value at the other receiving coil. The magnitude isoften referred to as the attenuation, and the phase is often called thephase shift. Thus, the magnitude, or attenuation, and the phase shift,or phase, are typically derived from the ratio of the voltage at onereceiver antenna relative to the voltage at another receiver antenna.

Commercially deployed MWD/LWD resistivity measurement systems usemultiple transmitters; consequently, attenuation and phase-basedresistivity values can be derived independently using each transmitteror from averages of signals from two or more transmitters. See forexample U.S. Pat. No. 5,594,343.

As demonstrated in U.S. Pat. Nos. 4,968,940 and 4,899,112, a very commonmethod practiced by those skilled in the art of MWD/LWD for determiningthe resistivity from the measured data is to transform the dielectricconstant into a variable that depends on the resistivity and then toindependently convert the phase shift and attenuation measurements totwo separate resistivity values. A key assumption implicitly used inthis practice is that each measurement senses the resistivity within thesame volume that it senses the dielectric constant. This implicitassumption is shown herein by the Applicant to be false. This currentlypracticed method may provide significantly incorrect resistivity values,even in virtually homogeneous earth formations; and the errors may beeven more severe in inhomogeneous formations.

A MWD/LWD tool typically transmits a 2 MHz signal (although frequenciesas low as 0.4 MHz are sometimes used). This frequency range is highenough to create difficulties in transforming the raw attenuation andphase measurements into accurate estimates of the resistivity and/or thedielectric constant. For example, the directly measured values are notlinearly dependent on either the resistivity or the dielectric constant(this nonlinearity, known to those skilled in the art as “skin-effect,”also limits the penetration of the fields into the earth formation). Inaddition, it is useful to separate the effects of the dielectricconstant and the resistivity on the attenuation and phase measurementsgiven that both the resistivity and the dielectric constant typicallyvary spatially within the earth formation. If the effects of both ofthese variables on the measurements are not separated, the estimate ofthe resistivity can be corrupted by the dielectric constant, and theestimate of the dielectric constant can be corrupted by the resistivity.Essentially, the utility of separating the effects is to obtainestimates of one parameter that do not depend on (are independent of)the other parameter. A commonly used current practice relies on assuminga correlative relationship between the resistivity and dielectricconstant (i.e., to transform the dielectric constant into a variablethat depends on the resistivity) and then calculating resistivity valuesindependently from the attenuation and phase shift measurements that areconsistent the correlative relationship. Differences between theresistivity values derived from corresponding phase and attenuationmeasurements are then ascribed to spatial variations (inhomogeneities)in the resistivity over the sensitive volume of the phase shift andattenuation measurements. See for example U.S. Pat. Nos. 4,899,112 and4,968,940. An implicit and instrumental assumption in this method isthat the attenuation measurement senses both the resistivity anddielectric constant within the same volume, and that the phase shiftmeasurement senses both variables within the same volume (but not thesame volume as the attenuation measurement). See for example U.S. Pat.Nos. 4,899,112 and 4,968,940. These assumptions facilitate theindependent determination of a resistivity value from a phasemeasurement and another resistivity value from an attenuationmeasurement. However, as is shown later, the implicit assumptionmentioned above is not true; so, the results determined using suchalgorithms are questionable. Certain methods are herein disclosed todetermine two resistivity values from a phase shift measurement and anattenuation measurement which do not use the false assumptions of theabove mentioned prior art.

Another method for determining the resistivity and/or dielectricconstant is to assume a model for the measurement apparatus in, forexample, a homogeneous medium (no spatial variation in either theresistivity or dielectric constant) and then to determine values for theresistivity and dielectric constant that cause the model to agree withthe measured phase shift and attenuation data. The resistivity anddielectric constant determined by the model are then correlated to theactual parameters of the earth formation. This method is thought to bevalid only in a homogeneous medium because of the implicit assumptionmentioned in the above paragraph. A recent publication by P. T. Wu, J.R. Lovell, B. Clark, S. D. Bonner, and J. R. Tabanou entitled“Dielectric-Independent 2-MHz Propagation Resistivities”(SPE 56448,1999) (hereafter referred to as “Wu”) demonstrates that such assumptionsare used by those skilled in the art. For example, Wu states: “Onefundamental assumption in the computation of Rex is an uninvadedhomogeneous formation. This is because the phase shift and attenuationinvestigate slightly different volumes.” It is shown herein by Applicantthat abandoning the false assumptions applied in this practice resultsin estimates of one parameter (i.e., the resistivity or dielectricconstant) that have no net sensitivity to the other parameter. Thisdesirable and previously unknown property of the results is very usefulbecause earth formations are commonly inhomogeneous.

Wireline dielectric measurement tools commonly use electrical signalshaving frequencies in the range 20 MHz-1.1 GHz. In this range, theskin-effect is even more severe, and it is even more useful to separatethe effects of the dielectric constant and resistivity. Those skilled inthe art of dielectric measurements have also falsely assumed that ameasurement (either attenuation or phase) senses both the resistivityand dielectric constant within the same volume. The design of themeasurement equipment and interpretation of the data both reflect this.See for example U.S. Pat. Nos. 4,185,238 and 4,209,747.

Wireline induction measurements are typically not attenuation and phase,but instead the real (R) and imaginary (X) parts of the voltage across areceiver antenna which consists of several induction coils in electricalseries. For the purpose of this disclosure, the R-signal for a wirelineinduction measurement corresponds to the phase measurement of a MWD/LWDresistivity or wireline dielectric tool, and the X-signal for a wirelineinduction measurement corresponds to the attenuation measurement of aMWD/LWD resistivity or wireline dielectric device. Wireline inductiontools typically operate using electrical signals at frequencies from8-200 kHz (most commonly at approximately 20 kHz). This frequency rangeis too low for significant dielectric sensitivity in normallyencountered cases; however, the skin-effect can corrupt the wirelineinduction measurements. As mentioned above, the skin-effect shows up asa non-linearity in the measurement as a function of the formationconductivity, and also as a dependence of the measurement sensitivityvalues on the formation conductivity. Estimates of the formationconductivity from wireline induction devices are often derived from dataprocessing algorithms which assume the tool response function is thesame at all depths within the processing window. The techniques of thisdisclosure can be applied to wireline induction measurements for thepurpose of deriving resistivity values without assuming the toolresponse function is the same at all depths within the processing windowas is done in U.S. Pat. No. 5,157,605. In order to make such anassumption, a background conductivity, σ, that applies for the datawithin the processing window is commonly used. Practicing a disclosedembodiment reduces the dependence of the results on the accuracy of theestimates for the background parameters because the backgroundparameters are not required to be the same at all depths within theprocessing window. In addition, practicing appropriate embodiments ofApplicant's techniques discussed herein reduces the need to performsteps to correct wireline induction data for the skin effect.

BRIEF SUMMARY OF THE INVENTION

Briefly, a resistivity measurement technique estimates a first value fora first electrical parameter consistent with an assumption that eachproperty of a measured electrical signal senses the first electricalparameter and a second electrical parameter in substantially the samevolume and estimates a second value of the first electrical parameterconsistent with the estimated first value and consistent with eachproperty of the measured electrical signal sensing the first electricalparameter and the second electrical parameter in different volumes. Thefirst parameter may be the resistivity of an earth formation, and thesecond parameter may be the dielectric constant of the earth formation.The measured electrical signal may include an attenuation measurementand a phase shift measurement. Applying this technique, a phaseconductivity may be determined from only a phase shift measurement. Anattenuation conductivity may then be determined based on an attenuationmeasurement and the phase conductivity. Since bandwidth is limited indata telemetry to a surface while drilling, phase shift measurements canbe telemetered without attenuation measurements for accomplishingresistivity measurements. Such results can be obtained more quickly byemploying a model of a hypothetical measurement device simpler than theactual measurement device.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

A better understanding of the present invention can be obtained when thefollowing detailed description of some preferred embodiments isconsidered in conjunction with the following drawings, in which:

FIG. 1 is a plot of multiple laboratory measurements on rock samplesrepresenting the relationship between the conductivity and thedielectric constant in a variety of geological media;

FIG. 2 illustrates the derivation of a sensitivity function in relationto an exemplary one-transmitter, one-receiver MWD/LWD resistivity tool;

FIG. 3 illustrates an exemplary one-transmitter, two-receiver MWD/LWDtool commonly referred to as an uncompensated measurement device;

FIG. 4 illustrates an exemplary two-transmitter, two-receiver MWD/LWDtool, commonly referred to as a compensated measurement device;

FIGS. 5a, 5 b, 5 c and 5 d are exemplary sensitivity function plots forDeep and Medium attenuation and phase shift measurements;

FIGS. 6a, 6 b, 6 c and 6 d are plots of the sensitivity functions forthe Deep and Medium measurements of 5 a, 5 b, 5 c and 5 d respectivelytransformed according to the techniques of a disclosed embodiment;

FIG. 7 is a portion of a table of background medium values and integralvalues employed in a disclosed embodiment;

FIG. 8 is a plot of attenuation and phase as a function of resistivityand dielectric constant;

FIG. 9 is a flowchart of a process that implements the techniques of adisclosed embodiment; and

FIG. 10 is a table of exemplary data values for a medium spaced 2 MHzmeasurement described in conjunction with FIG. 7.

DETAILED DESCRIPTION OF THE INVENTION

Some of the disclosed embodiments are relevant to both wirelineinduction and dielectric applications, as well asMeasurement-While-Drilling and Logging-While-Drilling (MWD/LWD)applications. Turning now to the figures, FIG. 1 is a plot ofmeasurements of the conductivity and dielectric constant determined bylaboratory measurements on a variety of rock samples from differentgeological environments. The points 121 through 129 represent measuredvalues of conductivity and dielectric constant (electrical parameters)for carbonate and sandstone earth formations. For instance, the point126 represents a sample with a conductivity value of 0.01 (10⁻²) siemensper meter (S/m) and a relative dielectric constant of approximately 22.It should be noted that both the conductivity scale and the dielectricscale are logarithmic scales; so, the data would appear to be much morescattered if they were plotted on linear scales.

The maximum boundary 111 indicates the maximum dielectric constantexpected to be observed at each corresponding conductivity. In a similarfashion, the minimum boundary 115 represents the minimum dielectricconstant expected to be observed at each corresponding conductivity. Thepoints 122 through 128 represent measured values that fall somewhere inbetween the minimum boundary 115 and the maximum boundary 111. A medianline 113 is a line drawn so that half the points, or points 121 through124 are below the median line 113 and half the points, or points 126through 129 are above the median line 113. The point 125 falls right ontop of the median line 113.

An elemental measurement between a single transmitting coil 205 and asingle receiving coil 207 is difficult to achieve in practice, but it isuseful for describing the sensitivity of the measurement to variationsof the conductivity and dielectric constant within a localized volume225 of an earth formation 215. FIG. 2 illustrates in more detailspecifically what is meant by the term “sensitivity function,” alsoreferred to as a “response function” or “geometrical factors.”Practitioners skilled in the art of wireline logging,Measurement-While-Drilling (MWD) and Logging-While-Drilling (LWD) arefamiliar with how to generalize the concept of a sensitivity function toapply to realistic measurements from devices using multiple transmittingand receiving antennas. Typically a MWD/LWD resistivity measurementdevice transmits a signal using a transmitter coil and measures thephase and magnitude of the signal at one receiver antenna 307 relativeto the values of the phase and the magnitude at another receiver antenna309 within a borehole 301 (FIG. 3). These relative values are commonlyreferred to as the phase shift and attenuation. It should be understoodthat one way to represent a complex signal with multiple components isas a phasor signal.

Sensitivity Functions

FIG. 2 illustrates an exemplary single transmitter, single receiverMWD/LWD resistivity tool 220 for investigating an earth formation 215. Ametal shaft, or “mandrel,” 203 is incorporated within the drill string(the drill string is not shown, but it is a series of pipes screwedtogether with a drill bit on the end), inserted into the borehole 201,and employed to take measurements of an electrical signal thatoriginates at a transmitter 205 and is sensed at a receiver 207. Themeasurement tool is usually not removed from the well until the drillstring is removed for the purpose of changing drill bits or becausedrilling is completed. Selected data from the tool are telemetered tothe surface while drilling. All data are typically recorded in memorybanks for retrieval after the tool is removed from the borehole 201.Devices with a single transmitter and a single receiver are usually notused in practice, but they are helpful for developing concepts such asthat of the sensitivity function. Schematic drawings of simple,practical apparatuses are shown in FIGS. 3 and 4.

In a wireline operation, the measurement apparatus is connected to acable (known as a wireline), lowered into the borehole 201, and data areacquired. This is done typically after the drilling operation isfinished. Wireline induction tools measure the real (R) and imaginary(X) components of the receiver 207 signal. The R and X-signalscorrespond to the phase shift and attenuation measurements respectively.In order to correlate the sensitivity of the phase shift and attenuationmeasurements to variations in the conductivity and dielectric constantof the earth formation 215 at different positions within the earthformation, the conductivity and dielectric constant within a smallvolume P 225 are varied. For simplicity, the volume P 225 is a solid ofrevolution about the tool axis (such a volume is called atwo-dimensional volume). The amount the phase and attenuationmeasurements change relative to the amount the conductivity anddielectric constant changed within P 225 is essentially the sensitivity.The sensitivity function primarily depends on the location of the pointP 225 relative to the locations of the transmitter 205 and receiver 207,on the properties of the earth formation 215, and on the excitationfrequency. It also depends on other variables such as the diameter andcomposition of the mandrel 203, especially when P is near the surface ofthe mandrel 203.

Although the analysis is carried out in two-dimensions, the importantconclusions regarding the sensitive volumes of phase shift andattenuation measurements with respect to the conductivity and dielectricconstant hold in three-dimensions. Consequently, the scope of thisapplication is not limited to two-dimensional cases. This is discussedmore in a subsequent section entitled, “ITERATIVE FORWARD MODELING ANDDIPPING BEDS.”

The sensitivity function can be represented as a complex number having areal and an imaginary part. In the notation used below, S, denotes acomplex sensitivity function, and its real part is S′, and its imaginarypart is S″. Thus, S=S′+iS″, in which the imaginary number i={square rootover (−1)}. The quantities S′ and S″ are commonly referred to asgeometrical factors or response functions. The volume P 225 is located adistance p in the radial direction from the tool's axis and a distance zin the axial direction from the receiver 206. S′ represents thesensitivity of attenuation to resistivity and the sensitivity of phaseshift to dielectric constant. Likewise, S″ represents the sensitivity ofattenuation to dielectric constant and the sensitivity of phase shift toresistivity. The width of the volume P 225 is Δρ 211 and the height ofthe volume P 225 is Δz 213. The quantity S′, or the sensitivity ofattenuation to resistivity, is calculated by determining the effect achange in the conductivity (reciprocal of resistivity) in volume P 225from a prescribed background value has on the attenuation of a signalbetween the transmitter 205 and the receiver 207, assuming thebackground conductivity value is otherwise unperturbed within the entireearth formation 215. In a similar fashion, S″, or the sensitivity of thephase to the resistivity, is calculated by determining the effect achange in the conductivity value in the volume P 225 from an assumedbackground conductivity value has on the phase of the signal between thetransmitter 205 and the receiver 207, assuming the background parametersare otherwise unperturbed within the earth formation 215. Alternatively,one could determine S′ and S″ by determining the effect a change in thedielectric constant within the volume P 225 has on the phase andattenuation, respectively. When the sensitivities are determined byconsidering a perturbation to the dielectric constant value within thevolume P 225, it is apparent that the sensitivity of the attenuation tochanges in the dielectric constant is the same as the sensitivity of thephase to the conductivity. It is also apparent that the sensitivity ofthe phase to the dielectric constant is the same as the sensitivity ofthe attenuation to the conductivity. By simultaneously considering thesensitivities of both the phase and attenuation measurement to thedielectric constant and to the conductivity, the Applicant shows apreviously unknown relationship between the attenuation and phase shiftmeasurements and the conductivity and dielectric constant values. Byemploying this previously unknown relationship, the Applicant providestechniques that produce better estimates of both the conductivity andthe dielectric constant values than was previously available from thosewith skill in the art. The sensitivity functions S′ and S″ and theirrelation to the subject matter of the Applicant's disclosure isexplained in more detail below in conjunction with FIGS. 5a-d and FIGS.6a-d.

In the above, sensitivities to the dielectric constant were referred to.Strictly speaking, the sensitivity to the radian frequency) times thedielectric constant should have been referred to. This distinction istrivial to those skilled in the art.

In FIG. 2, if the background conductivity (reciprocal of resistivity) ofthe earth formation 215 is σ₀ and the background dielectric constant ofthe earth formation 215 is ε₀, then the ratio of the receiver 207voltage to the transmitter 205 current in the background medium can beexpressed as Z_(RT) ⁰, where R stands for the receiver 207 and T standsfor the transmitter 205. Hereafter, a numbered subscript or superscriptsuch as the ‘0’ is merely used to identify a specific incidence of thecorresponding variable or function. If an exponent is used, the variableor function being raised to the power indicated by the exponent will besurrounded by parentheses and the exponent will be placed outside theparentheses. For example (L₁)³ would represent the variable L₁, raisedto the third power.

When the background conductivity σ₀, and/or dielectric constant ε₀ arereplaced new values σ₁, and/or ε₁ in the volume P 225, the ratio betweenthe receiver 207 voltage to the transmitter 205 current is representedby Z_(RT) ¹. Using the same nomenclature, a ratio between a voltage at ahypothetical receiver placed in the volume P 225 and the current at thetransmitter 205 can be expressed as Z_(PT) ⁰. In addition, a ratiobetween the voltage at the receiver 207 and a current at a hypotheticaltransmitter in the volume P 225 can be expressed as Z_(RP) ⁰. Using theBorn approximation, it can be shown that,$\frac{Z_{RT}^{1}}{Z_{RT}^{0}} = {1 + {{S( {T,R,P} )}\Delta \quad \overset{\sim}{\sigma}\Delta \quad \rho \quad \Delta \quad z}}$

where the sensitivity function, defined as S(TR,P), is${S( {T,R,P} )} = {- \frac{Z_{RP}^{0}Z_{PT}^{0}}{2\pi \quad \rho \quad Z_{RT}^{0}}}$

in which Δ{tilde over (σ)}={tilde over (σ)}₁−{tilde over(σ)}₀=(σ₁−σ₀)+iω(ε₁−ε₀), and the radian frequency of the transmittercurrent is ω=2πf. A measurement of this type, in which there is just onetransmitter 205 and one receiver 207, is defined as an “elemental”measurement. It should be noted that the above result is also valid ifthe background medium parameters vary spatially within the earthformation 215. In the above equations, both the sensitivity functionS(T,R,P) and the perturbation Δ{tilde over (σ)} are complex-valued. Somedisclosed embodiments consistently treat the measurements, theirsensitivities, and the parameters to be estimated as complex-valuedfunctions. This is not done in the prior art.

The above sensitivity function of the form S(T,R,P) is referred to as a2-D (or two-dimensional) sensitivity function because the volume ΔρΔzsurrounding the point P 225, is a solid of revolution about the axis ofthe tool 201. Because the Born approximation was used, the sensitivityfunction S depends only on the properties of the background mediumbecause it is assumed that the same field is incident on the pointP(ρ,z) even though the background parameters have been replaced by{tilde over (σ)}₁.

FIG. 3 illustrates an exemplary one-transmitter, two-receiver MWD/LWDresistivity measurement apparatus 320 for investigating an earthformation 315. Due to its configuration, the tool 320 is defined as an“uncompensated” device and collects uncompensated measurements from theearth formation 315. For the sake of simplicity, a borehole is notshown. This measurement tool 320 includes a transmitter 305 and tworeceivers 307 and 309, each of which is incorporated into a metalmandrel 303. Typically, the measurement made by such a device is theratio of the voltages at receivers 307 and 309. In this example, usingthe notation described above in conjunction with FIG. 2, the sensitivityfunction S(T,R,R′,P) for the uncompensated device can be shown to be thedifference between the elemental sensitivity functions S(T,R,P) andS(T,R′,P), where T represents the transmitter 305, R represents thereceiver 307, R′ represents the receiver 309, and P represents a volume(not shown) similar to the volume P 225 of FIG. 2.

For wireline induction measurements, the voltage at the receiver R issubtracted from the voltage at the receiver R′, and the position andnumber of turns of wire for R are commonly chosen so that the differencein the voltages at the two receiver antennas is zero when the tool is ina nonconductive medium. For MWD/LWD resistivity and wireline dielectricconstant measurements, the voltage at the receiver R, or V_(R), and thevoltage at the receiver R′, or V_(R′), are examined as the ratioV_(R)/V_(R′). In either case, it can be shown that

S(T,R,R′,P)=S(T,R,P)−S(T,R′,P).

The sensitivity for an uncompensated measurement is the differencebetween the sensitivities of two elemental measurements such as S(T,R,P)and S(T,R′,P) calculated as described above in conjunction with FIG. 2.

FIG. 4 illustrates an exemplary two-transmitter, two-receiver MWD/LWDresistivity tool 420. Due to its configuration (transmitters beingdisposed symmetrically), the tool 420 is defined as a “compensated” tooland collects compensated measurements from an earth formation 415. Thetool 420 includes two transmitters 405 and 411 and two receivers 407 and409, each of which is incorporated into a metal mandrel or collar 403.Each compensated measurement is the geometric mean of two correspondinguncompensated measurements. In other words, during a particulartimeframe, the tool 420 performs two uncompensated measurements, oneemploying transmitter 405 and the receivers 407 and 409 and the otheremploying the transmitter 411 and the receivers 409 and 407. These twouncompensated measurements are similar to the uncompensated measurementdescribed above in conjunction with FIG. 3. The sensitivity function Sof the tool 420 is then defined as the arithmetic average of thesensitivity functions for each of the uncompensated measurements.Another way to describe this relationship is with the following formula:${S( {T,R,R^{\prime},T^{\prime},P} )} = {\frac{1}{2}\lbrack {{S( {T,R,R^{\prime},P} )} + {S( {T^{\prime},R^{\prime},R,P} )}} \rbrack}$

where T represents transmitter 405, T′ represents transmitter 411, Rrepresents receiver 407, R′ represents receiver 409 and P represents asmall volume of the earth formation similar to 225 (FIG. 2).

The techniques of the disclosed embodiments are explained in terms of acompensated tool such as the tool 420 and compensated measurements suchas those described in conjunction with FIG. 4. However, it should beunderstood that the techniques also apply to uncompensated tools such asthe tool 320 and uncompensated measurements described above inconjunction with FIG. 3 and elemental tools such as the tool 220 andelemental measurements such as those described above in conjunction withFIG. 2. In addition, the techniques are applicable for use in a wirelinesystem, a system that may not incorporate its transmitters and receiversinto a metal mandrel, but may rather affix a transmitter and a receiverto a tool made of a non-conducting material such as fiberglass. Thewireline induction frequency is typically too low for dielectric effectsto be significant. Also typical for wireline induction systems is toselect the position and number of turns of groups of receiver antennasso that there is a null signal in a nonconductive medium. When this isdone, Z_(RT) ⁰=0 if {tilde over (σ)}₀=0. As a result, it is necessary tomultiply the sensitivity and other quantities by Z_(RT) ⁰ to use theformulation given here in such cases.

The quantity Z_(RT) ¹/Z_(RT) ⁰ can be expressed as a complex numberwhich has both a magnitude and a phase (or alternatively real andimaginary parts). To a good approximation, the raw attenuation value(which corresponds to the magnitude) is: $\begin{matrix}{{{\frac{Z_{RT}^{1}}{Z_{RT}^{0}}} \approx \quad {{Re}\lbrack \frac{Z_{RT}^{1}}{Z_{RT}^{0}} \rbrack}} = {1 + {{{Re}\lbrack {{S( {T,R,P} )}\Delta \quad \overset{\sim}{\sigma}} \rbrack}\Delta \quad \rho \quad \Delta \quad z}}} \\{= \quad {1 + {\lbrack {{S^{\prime}{\Delta\sigma}} - {S^{''}\omega \quad \Delta \quad ɛ}} \rbrack \Delta \quad \rho \quad \Delta \quad z}}}\end{matrix}$

where the function Re[.] denotes the real part of its argument. Also, toa good approximation, the raw phase shift value is: $\begin{matrix}{{{{phase}( \frac{Z_{RT}^{1}}{Z_{RT}^{0}} )} \approx \quad {{Im}\lbrack \frac{Z_{RT}^{1}}{Z_{RT}^{0}} \rbrack}} = {{{Im}\lbrack {{S( {T,R,P} )}\Delta \quad \overset{\sim}{\sigma}} \rbrack}\Delta \quad \rho \quad \Delta \quad z}} \\{= \quad {\lbrack {{S^{''}\Delta \quad \sigma} + {S^{\prime}{\omega\Delta}\quad ɛ}} \rbrack {\Delta\rho\Delta}\quad z}}\end{matrix}$

in which Im[.] denotes the imaginary part of its argument,S(T,R,P)=S′+iS″, Δσ=σ₁−σ₀, and Δε=ε₁−ε₀. For the attenuationmeasurement, S′ is the sensitivity to the resistivity and S″ is thesensitivity to the dielectric constant. For the phase shift measurement,S′ is the sensitivity to the dielectric constant and S″ is thesensitivity to the resistivity. This is apparent because S′ is thecoefficient of Δσ in the equation for attenuation, and it is also thecoefficient of ωΔε in the equation for the phase shift. Similarly, S″ isthe coefficient of Δσ in the equation for the phase shift, and it isalso the coefficient for −ωΔε in the equation for attenuation. Thisimplies that the attenuation measurement senses the resistivity in thesame volume as the phase shift measurement senses the dielectricconstant and that the phase shift measurement senses the resistivity inthe same volume as the attenuation measurement senses the dielectricconstant. In the above, we have referred to sensitivities to thedielectric constant. Strictly speaking, the sensitivity to the radianfrequency ω times the dielectric constant Δε should have been referredto. This distinction is trivial to those skilled in the art.

The above conclusion regarding the volumes in which phase andattenuation measurements sense the resistivity and dielectric constantfrom Applicant's derived equations also follows from a well known resultfrom complex variable theory known in that art as the Cauchy-Reimannequations. These equations provide the relationship between thederivatives of the real and imaginary parts of an analytic complexfunction with respect to the real and imaginary parts of the function'sargument.

FIGS. 5a, 5 b, 5 c and 5 d can best be described and understoodtogether. In all cases, the mandrel diameter is 6.75 inches, thetransmitter frequency is 2 MHz, and the background medium ischaracterized by a conductivity of σ₀=0.01 S/m and a relative dielectricconstant of ε₀=10. The data in FIGS. 5a and 5 c labeled “MediumMeasurement” are for a compensated type of design shown in FIG. 4. Theexemplary distances between transmitter 405 and receivers 407 and 409are 20 and 30 inches, respectively. Since the tool is symmetric, thedistances between transmitter 411 and receivers 409 and 407 are 20 and30 inches, respectively. The data in FIGS. 5b and 5 d labeled “DeepMeasurement” are also for a compensated tool as shown in FIG. 4, butwith exemplary transmitter-receiver spacings of 50 and 60 inches. Eachplot shows the sensitivity of a given measurement as a function ofposition within the formation. The term sensitive volume refers to theshape of each plot as well as its value at any point in the formation.The axes labeled “Axial Distance” refer to the coordinate along the axisof the tool with zero being the geometric mid-point of the antenna array(halfway between receivers 407 and 409) to a given point in theformation. The axes labeled “Radial Distance” refer to the radialdistance from the axis of the tool to a given point in the formation.The value on the vertical axis is actually the sensitivity value for theindicated measurement. Thus, FIG. 5a is a plot of a sensitivity functionthat illustrates the sensitivity of the “Medium” phase shift measurementin relation to changes in the resistivity as a function of the locationof the point P 225 in the earth formation 215 (FIG. 2). If themeasurement of phase shift changes significantly in response to changingthe resistivity from its background value, then phase shift isconsidered relatively sensitive to the resistivity at the point P 225.If the measurement of phase shift does not change significantly inresponse to changing the resistivity, then the phase shift is consideredrelatively insensitive at the point P 225. Based upon the relationshipdisclosed herein, FIG. 5a also illustrates the sensitivity of the“Medium” attenuation measurement in relation to changes in dielectricconstant values. Note that the dimensions of the sensitivity on thevertical axes is ohms per meter (Ω/m) and distances on the horizontalaxes are listed in inches. In a similar fashion, FIG. 5b is a plot ofthe sensitivity of the attenuation measurement to the resistivity. Basedon the relationship disclosed herein, FIG. 5b is also the sensitivity ofa phase shift measurement to a change in the dielectric constant. FIGS.5b and 5 d have the same descriptions as FIGS. 5a and 5 c, respectively,but FIGS. 5b and 5 d are for the “Deep Measurement” with the antennaspacings described above.

Note that the shape of FIG. 5a is very dissimilar to the shape of FIG.5c. This means that the underlying measurements are sensitive to thevariables in different volumes. For example, the Medium phase shiftmeasurement has a sensitive volume characterized by FIG. 5a for theresistivity, but this measurement has the sensitive volume shown in FIG.5c for the dielectric constant. FIGS. 5a-5 d illustrate that for aparticular measurement the surface S″ is more localized than the surfaceS′ such that the sensitive volume associated with the surface S′substantially encloses the sensitive volume associated with S″ for thecorresponding measurement. As discussed below, it is possible totransform an attenuation and a phase shift measurement to a complexnumber which has the following desirable properties: 1) its real part issensitive to the resistivity in the same volume that the imaginary partis sensitive to the dielectric constant; 2) the real part has no netsensitivity to the dielectric constant; and, 3) the imaginary part hasno net sensitivity to the resistivity. In addition, the transformationis generalized to accommodate multiple measurements acquired at multipledepths. The generalized method can be used to produce independentestimates of the resistivity and dielectric constant within a pluralityof volumes within the earth formation.

Transformed Sensitivity Functions and Transformation of the Measurements

For simplicity, the phase shift and attenuation will not be used.Hereafter, the real and imaginary parts of measurement will be referredto instead. Thus,

w=w′+iw″

w′=(10)^(dB/20)×cos(θ)

w″=(10)^(dB/20)×sin(θ)

where w′ is the real part of w, w″ is the imaginary part of w, i is thesquare root of the integer −1, dB is the attenuation in decibels, and θis the phase shift in radians.

The equations that follow can be related to the sensitivity functionsdescribed above in conjunction with FIG. 2 by defining variablesw₁=Z_(RT) ¹ and w₀=Z_(RT) ⁰. The variable w₁ denotes an actual toolmeasurement in the earth formation 215. The variable w₀ denotes theexpected value for the tool measurement in the background earthformation 215. For realistic measurement devices such as those describedin FIGS. 3 and 4, the values for w₁ and w₀ would be the voltage ratiosdefined in the detailed description of FIGS. 3 and 4. In one embodiment,the parameters for the background medium are determined and then used tocalculate value of w₀ using a mathematical model to evaluate the toolresponse in the background medium. One of many alternative methods todetermine the background medium parameters is to estimate w₀ directlyfrom the measurements, and then to determine the background parametersby correlating w₀ to a model of the tool in the formation which has thebackground parameters as inputs.

As explained in conjunction with FIG. 2, the sensitivity function Srelates the change in the measurement to a change in the mediumparameters such as resistivity and dielectric constant within a smallvolume 225 of the earth formation 215 at a prescribed location in theearth formation 225, or background medium. A change in measurements dueto small variations in the medium parameters at a range of locations canbe calculated by integrating the responses from each such volume in theearth formation 215. Thus, if Δ{tilde over (σ)} is defined for a largenumber of points ρ,z, then

Z _(RT) ¹ =R _(RT) ⁰(1+I[SΔ{tilde over (σ)}])

in which I is a spatial integral function further defined asI[F] = ∫_(−∞)^(+∞)z∫₀^(+∞)ρ  F(ρ, z)

where F is a complex function.

Although the perturbation from the background medium, Δ{tilde over (σ)}is a function of position, parameters of a hypothetical, equivalenthomogeneous perturbation (meaning that no spatial variations are assumedin the difference between the resistivity and dielectric constant andvalues for both of these parameters in the background medium) can bedetermined by assuming the perturbation is not a function of positionand then solving for it. Thus,

Δ{circumflex over (σ)}I[S]=I[SΔ{tilde over (σ)}]

where Δ{circumflex over (σ)} represents the parameters of the equivalenthomogeneous perturbation. From the previous equations, it is clear that${\Delta \quad \hat{\sigma}} = {\frac{I\lbrack {S\quad \Delta \quad \overset{\sim}{\sigma}} \rbrack}{I\lbrack S\rbrack} = {{I\lbrack {\hat{S}\Delta \quad \overset{\sim}{\sigma}} \rbrack} = {\frac{1}{I\lbrack S\rbrack}( {\frac{w_{1}}{w_{0}} - 1} )}}}$

and $\hat{S} = \frac{S}{I\lbrack S\rbrack}$

where Δ{circumflex over (σ)} is the transformed measurement (it isunderstood that Δ{circumflex over (σ)} is also the equivalenthomogeneous perturbation and that the terms transformed measurement andequivalent homogeneous perturbation will be used synonymously), Ŝ is thesensitivity function for the transformed measurement, and Ŝ will bereferred to as the transformed sensitivity function. In the above, w₁ isthe actual measurement, and w₀ is the value assumed by the measurementin the background medium. An analysis of the transformed sensitivityfunction Ŝ, shows that the transformed measurements have the followingproperties: 1) the real part of Δ{circumflex over (σ)} is sensitive tothe resistivity in the same volume that its imaginary part is sensitiveto the dielectric constant; 2) the real part of Δ{circumflex over (σ)}as has no net sensitivity to the dielectric constant; and, 3) theimaginary part of Δ{circumflex over (σ)} has no net sensitivity to theresistivity. Details of this analysis will be given in the next fewparagraphs.

The techniques of the disclosed embodiment can be further refined byintroducing a calibration factor c (which is generally a complex numberthat may depend on the temperature of the measurement apparatus andother environmental variables) to adjust for anomalies in the physicalmeasurement apparatus. In addition, the term, W_(bh) can be introducedto adjust for effects caused by the borehole 201 on the measurement.With these modifications, the transformation equation becomes${\Delta \quad \hat{\sigma}} = {\frac{I\lbrack {S\quad \Delta \quad \overset{\sim}{\sigma}} \rbrack}{I\quad\lbrack S\rbrack} = {{I\lbrack {\hat{S}\quad \Delta \quad \overset{\sim}{\sigma}} \rbrack} = {\frac{1}{I\lbrack S\rbrack}{( {\frac{{cw}_{1} - w_{bh}}{w_{0}} - 1} ).}}}}$

The sensitivity function for the transformed measurement is determinedby applying the transformation to the original sensitivity function, S.Thus,$\hat{S} = {{{\hat{S}}^{\prime} + {i{\hat{S}}^{''}}} = {\frac{S}{I\lbrack S\rbrack} = {\frac{{S^{\prime}{I\lbrack S^{\prime} \rbrack}} + {S^{''}{I\lbrack S^{''} \rbrack}}}{{{I\lbrack S\rbrack}}^{2}} + {i{\frac{{S^{''}{I\lbrack S^{\prime} \rbrack}} - {S^{\prime}{I\lbrack S^{''} \rbrack}}}{{{I\lbrack S\rbrack}}^{2}}.}}}}}$

Note that I[Ŝ]=I[Ŝ′]=1 because I[Ŝ″]=0. The parameters for theequivalent homogeneous perturbation are

Δ{circumflex over (σ)}′={circumflex over (σ)}₁−σ₀ =I[Ŝ′Δσ]−I[Ŝ″ωΔε]

Δ{circumflex over (σ)}″=ω({circumflex over (ε)}₁−ε₀)=I[Ŝ′ωΔε]+I[Ŝ″Δσ].

The estimate for the conductivity perturbation, Δ{circumflex over (σ)}′suppresses sensitivity (is relatively insensitive) to the dielectricconstant perturbation, and the estimate of the dielectric constantperturbation, Δ{circumflex over (σ)}″/ω suppresses sensitivity to theconductivity perturbation. This is apparent because the coefficient ofthe suppressed variable is Ŝ″. In fact, the estimate for theconductivity perturbation Δ{circumflex over (σ)}″ is independent of thedielectric constant perturbation provided that deviations in thedielectric constant from its background are such that I[Ŝ″ωΔε]=0. SinceI[Ŝ″]=0, this is apparently the case if ωΔε is independent of position.Likewise, the estimate for the dielectric constant perturbation given byΔ{circumflex over (σ)}″/ω is independent of the conductivityperturbation provided that deviations in the conductivity from itsbackground value are such that I[Ŝ″Δσ]=0. Since I[Ŝ″]=0, this isapparently the case if Δσ is independent of position.

Turning now to FIGS. 6a and 6 b, illustrated are plots of thesensitivity functions Ŝ′ and Ŝ″ derived from S′ and S″ for the mediumtransmitter-receiver spacing measurement shown in FIGS. 5a and 5 c usingthe transformation $\hat{S} = {\frac{S}{I\lbrack S\rbrack}.}$

The data in FIGS. 6c and 6 d were derived from the data in FIGS. 5b and5 d for the Deep T-R spacing measurement. As shown in FIGS. 6a, 6 b, 6 cand 6 d, using the transformed measurements to determine the electricalparameters of the earth formation is a substantial improvement over theprior art. The estimates of the medium parameters are more accurate andless susceptible to errors in the estimate of the background mediumbecause the calculation of the resistivity is relatively unaffected bythe dielectric constant and the calculation of the dielectric constantis relatively unaffected by the resistivity. In addition to integratingto 0, the peak values for Ŝ″ in FIGS. 6b and 6 d are significantly lessthan the respective peak values for Ŝ′ in FIGS. 6a and 6 c. Both ofthese properties are very desirable because Ŝ″ is the sensitivityfunction for the variable that is suppressed.

Realization of the Transformation

In order to realize the transformation, it is desirable to have valuesof I[S] readily accessible over the range of background mediumparameters that will be encountered. One way to achieve this is tocompute the values for I[S] and then store them in a lookup table foruse later. Of course, it is not necessary to store these data in such alookup table if it is practical to quickly calculate the values for I[S]on command when they are needed. In general, the values for I[S] can becomputed by directly; however, it can be shown that${{{I\lbrack S\rbrack} = {\frac{1}{w_{0}}\frac{\partial w}{\partial\overset{\sim}{\sigma}}}}}_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}}$

where w₀ is the expected value for the measurement in the backgroundmedium, and the indicated derivative is calculated using the followingdefinition:${\frac{\partial w}{\partial\overset{\sim}{\sigma}}}_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}} = {\lim\limits_{{\Delta \quad \overset{\sim}{\sigma}}arrow 0}{\frac{{w( {{\overset{\sim}{\sigma}}_{0} + {\Delta \quad \overset{\sim}{\sigma}}} )} - {w( {\overset{\sim}{\sigma}}_{0} )}}{( {{\overset{\sim}{\sigma}}_{0} + {\Delta \quad \overset{\sim}{\sigma}}} ) - {\overset{\sim}{\sigma}}_{0}}.}}$

In the above formula, {tilde over (σ)}₀ may vary from point to point inthe formation 215 (the background medium may be inhomogeneous), but theperturbation Δ{tilde over (σ)} is constant at all points in theformation 215. As an example of evaluating I[S] using the above formula,consider the idealized case of a homogeneous medium with a smalltransmitter coil and two receiver coils spaced a distance L₁ and L₂ fromthe transmitter. Then,${{{w_{0} = {( \frac{L_{1}}{L_{2}} )^{3}\frac{{\exp ( {{ik}_{0}L_{2}} )}( {1 - {{ik}_{0}L_{2}}} )}{{\exp ( {{ik}_{0}L_{1}} )}( {1 - {{ik}_{0}L_{1}}} )}}}{{I\lbrack S\rbrack} = {\frac{1}{w_{0}}\frac{\partial w}{\partial\overset{\sim}{\sigma}}}}}}_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}} = {\frac{i\quad {\omega\mu}}{2}( {\frac{( L_{2} )^{2}}{1 - {{ik}_{0}L_{2}}} - \frac{( L_{1} )^{2}}{1 - {{ik}_{0}L_{1}}}} )}$

The wave number in the background medium is k₀={square root over(iωμ{tilde over (σ)})}₀, the function exp(.) is the complex exponentialfunction where exp(1)≈2.71828, and the symbol μ denotes the magneticpermeability of the earth formation. The above formula for I[S] appliesto both uncompensated (FIG. 3) and to compensated (FIG. 4) measurementsbecause the background medium has reflection symmetry about the centerof the antenna array in FIG. 4.

For the purpose of this example, the above formula is used to computethe values for I[S]=I[S′]+iI[S″]. FIG. 7 illustrates an exemplary table701 employed in a Create Lookup Table step 903 (FIG. 9) of the techniqueof the disclosed embodiment. Step 903 generates a table such as table701 including values for the integral of the sensitivity function overthe range of variables of interest. The first two columns of the table701 represent the conductivity σ₀ and the dielectric constant ε₀ of thebackground medium. The third and fourth columns of the table 701represent calculated values for the functions I[S′] and I[S″] for a Deepmeasurement, in which the spacing between the transmitter 305 receivers307 and 309 is 50 and 60 inches, respectively. The fifth and sixthcolumns of the table 701 represent calculated values for the functionsI[S′] and I[S″] for a Medium measurement, in which the spacing betweenthe transmitter 305 receivers 307 and 309 is 20 and 30 inches,respectively. It is understood that both the frequency of thetransmitter(s) and the spacing between the transmitter(s) andreceiver(s) can be varied. Based upon this disclosure, it is readilyapparent to those skilled in the art that algorithms such as the onedescribed above can be applied to alternative measurementconfigurations. If more complicated background media are used, forexample including the mandrel with finite-diameter antennas, it may bemore practical to form a large lookup table such as table 701 but withmany more values. Instead of calculating I[S] every time a value isneeded, data would be interpolated from the table. Nonetheless, table701 clearly illustrates the nature of such a lookup table. Such a tablewould contain the values of the functions I[S′] and I[S″] for the entirerange of values of the conductivity so, and the dielectric constant ε₀likely to be encountered in typical earth formations. For example, I[S′]and I[S″] could be calculated for values of ε₀ between 1 and 1000 andfor values of σ₀ between 0.0001 and 10.0. Whether calculating values forthe entire lookup table 701 or computing the I[S′] and I[S″] on commandas needed, the data is used as explained below.

FIG. 8 illustrates a chart 801 used to implement a Determine BackgroundMedium Parameters step 905 (FIG. 9) of the techniques of the disclosedembodiment. The chart 801 represents a plot of the attenuation and phaseshift as a function of resistivity and dielectric constant in ahomogeneous medium. Similar plots can be derived for more complicatedmedia. However, the homogeneous background media are routinely used dueto their simplicity. Well known numerical methods such as inverseinterpolation can be used to calculate an initial estimate of backgroundparameters based upon the chart 801. In one embodiment, the measuredattenuation and phase shift values are averaged over a few feet of depthwithin the borehole 201. These average values are used to determine thebackground resistivity and dielectric constant based upon the chart 801.It should be understood that background medium parameters can beestimated in a variety of ways using one or more attenuation and phasemeasurements.

FIG. 9 is a flowchart of an embodiment of the disclosed transformationtechniques that can be implemented in a software program which isexecuted by a processor of a computing system such as a computer at thesurface or a “downhole” microprocessor. Starting in a Begin Analysisstep 901, control proceeds immediately to the Create Lookup Table step903 described above in conjunction with FIG. 7. In an alternativeembodiment, step 903 can be bypassed and the function of the lookuptable replaced by curve matching, or “forward modeling.” Control thenproceeds to an Acquire Measured Data Step 904. Next, control proceeds toa Determine Background Values Step 905, in which the background valuesfor the background medium are determined. Step 905 corresponds to thechart 801 (FIG. 8).

Control then proceeds to a Determine Integral Value step 907. TheDetermine Integral Value step 907 of the disclosed embodiment determinesan appropriate value for I[S] using the lookup table generated in thestep 903 described above or by directly calculating the I[S] value asdescribed in conjunction with FIG. 7. Compute Parameter Estimate, step909, computes an estimate for the conductivity and dielectric constantas described above using the following equation:${\Delta \quad \hat{\sigma}} = {\frac{I\lbrack {S\quad \Delta \quad \overset{\sim}{\sigma}} \rbrack}{I\lbrack S\rbrack} = {{I\lbrack {\hat{S}\Delta \quad \overset{\sim}{\sigma}} \rbrack} = {\frac{1}{I\lbrack S\rbrack}{( {\frac{{cw}_{1} - w_{bh}}{w_{0}} - 1} ).}}}}$

where the borehole effect and a calibration factor are taken intoaccount using the factors w_(bh) and c, respectively. The conductivityvalue plotted on the log (this is the value correlated to theconductivity of the actual earth formation) is Re(Δ{circumflex over(σ)}+{circumflex over (σ)}₀) where the background medium ischaracterized by {circumflex over (σ)}₀. The estimate for the dielectricconstant can also be plotted on the log (this value is correlated to thedielectric constant of the earth formation), and this value isIm(Δ{circumflex over (σ)}+{tilde over (σ)}₀)/ω. Lastly, in the FinalDepth step 911, it is determined whether the tool 201 is at the finaldepth within the earth formation 215 that will be considered in thecurrent logging pass. If the answer is “Yes,” then control proceeds to astep 921 where is processing is complete. If the answer in step 911 is“No,” control proceeds to a Increment Depth step 913 where the tool 220is moved to its next position in the borehole 201 which penetrates theearth formation 215. After incrementing the depth of the tool 220,control proceeds to step 904 where the process of steps 904, 905, 907,909 and 911 are repeated. It should be understood by those skilled inthe art that embodiments described herein in the form a computing systemor as a programmed electrical circuit can be realized.

Improved estimates for the conductivity and/or dielectric constant canbe determined by simultaneously considering multiple measurements atmultiple depths. This procedure is described in more detail below underthe heading “Multiple Sensors At Multiple Depths.”

Multiple Sensors at Multiple Depths

In the embodiments described above, the simplifying assumption thatΔ{tilde over (σ)} is not position dependent facilitates determining avalue for Δ{circumflex over (σ)} associated with each measurement byconsidering only that measurement at a single depth within the well (atleast given a background value {tilde over (σ)}₀). It is possible toeliminate the assumption that Δ{tilde over (σ)} is independent ofposition by considering data at multiple depths, and in general, to alsoconsider multiple measurements at each depth. An embodiment of such atechnique for jointly transforming data from multiple MWD/LWD sensors atmultiple depths is given below. Such an embodiment can also be used forprocessing data from a wireline dielectric tool or a wireline inductiontool. Alternate embodiments can be developed based on the teachings ofthis disclosure by those skilled in the art.

In the disclosed example, the background medium is not assumed to be thesame at all depths within the processing window. In cases where it ispossible to assume the background medium is the same at all depthswithin the processing window, the system of equations to be solved is inthe form of a convolution. The solution to such systems of equations canbe expressed as a weighted sum of the measurements, and the weights canbe determined using standard numerical methods. Such means are known tothose skilled in the art, and are referred to as “deconvolution”techniques. It will be readily understood by those with skill in the artthat deconvolution techniques can be practiced in conjunction with thedisclosed embodiments without departing from the spirit of theinvention, but that the attendant assumptions are not necessary topractice the disclosed embodiments in general.

Devices operating at multiple frequencies are considered below, butmultifrequency operation is not necessary to practice the disclosedembodiments. Due to frequency dispersion (i.e., frequency dependence ofthe dielectric constant and/or the conductivity value), it is notnecessarily preferable to operate using multiple frequencies. Given thedisclosed embodiments, it is actually possible to determine thedielectric constant and resistivity from single-frequency data. In fact,the disclosed embodiments can be used to determine and quantifydispersion by separately processing data sets acquired at differentfrequencies. In the below discussion, it is understood that subsets ofdata from a given measurement apparatus or even from several apparatusescan be processed independently to determine parameters of interest. Thebelow disclosed embodiment is based on using all the data availablestrictly for purpose of simplifying the discussion.

Suppose multiple transmitter-receiver spacings are used and that eachtransmitter is excited using one or more frequencies. Further, supposedata are collected at multiple depths in the earth formation 215. Let Ndenote the number of independent measurements performed at each ofseveral depths, where a measurement is defined as the data acquired at aparticular frequency from a particular set of transmitters and receiversas shown in FIGS. 3 or 4. Then, at each depth z_(k), a vector of all themeasurements can be defined as${{\overset{\_}{v}}_{k}\lbrack {( {\frac{w_{1}}{w_{0}} - 1} )_{k1},( {\frac{w_{1}}{w_{0}} - 1} )_{k2},\ldots \quad,( {\frac{w_{1}}{w_{0}} - 1} )_{kN}} \rbrack}^{T},$

and the perturbation of the medium parameters from the background mediumvalues associated with these measurements is

Δ{tilde over ({overscore (σ)})}=Δ{tilde over (σ)}(ρ,z)[1,1, . . .,1]^(T)

in which the superscript T denotes a matrix transpose, {overscore(v)}_(k) is a vector each element of which is a measurement, and Δ{tildeover ({overscore (σ)})} is a vector each element of which is aperturbation from the background medium associated with a correspondingelement of {overscore (v)}_(k) at the point P 225. In the above, thedependence of the perturbation, Δ{tilde over (σ)}(ρ,z) on the positionof the point P 225 is explicitly denoted by the variables p and z. Ingeneral, the conductivity and dielectric constant of both the backgroundmedium and the perturbed medium depend on p and z; consequently, nosubscript k needs to be associated with Δ{tilde over (σ)}(ρ,z), and allelements of the vector Δ{tilde over ({overscore (σ)})} are equal. Asdescribed above, borehole corrections and a calibration can be appliedto each measurement, but here they are omitted for simplicity.

The vectors {overscore (v)}_(k) and Δ{tilde over ({overscore (σ)})} arerelated as follows:

{overscore (v)}_(k) =I[{double overscore (S)}Δ{tilde over ({overscore(σ)})}]

in which {double overscore (S)} is a diagonal matrix with each diagonalelement being the sensitivity function centered on the depth z_(k), forthe corresponding element of {overscore (v)}_(k), and the integraloperator I is defined by: I[F] = ∫_(−∞)^(+∞)z∫₀^(+∞)ρ  F(ρ, z).

Using the notationI_(mn)[F] = ∫_(z_(m − 1))^(z_(m))z∫_(ρ_(n − 1))^(ρ_(n))ρ  F(ρ, z)

to denote integrals of a function over the indicated limits ofintegration, it is apparent that${\overset{\_}{v}}_{k} = {\sum\limits_{m = {- M}}^{+ M}{\sum\limits_{n = 1}^{N^{\prime}}{I_{mn}\lbrack {\overset{\_}{\overset{\_}{S}}\Delta \quad \overset{\_}{\overset{\sim}{\sigma}}} \rbrack}}}$

if ρ₀=0, ρ_(N′)=+∞, z_(−M−1)=−∞, and z_(M)=+∞. The equation directlyabove is an integral equation from which an estimate of Δ{tilde over(σ)}(ρ,z) can be calculated. With the definitions ρ_(n)^(*)=(ρ_(n)+ρ_(n−1))/2 and z_(m) ^(*)=(z_(m)+z_(m−1))/2 and making theapproximation Δ{tilde over (σ)}(ρ,z)=Δ{circumflex over (σ)}(ρ_(n)^(*),z_(m) ^(*)) within the volumes associated with each value for m andn, it follows that${\overset{\_}{v}}_{k} = {\sum\limits_{m = {- M}}^{+ M}{\sum\limits_{n = 1}^{N^{\prime}}{{I_{mn}\lbrack \overset{\_}{\overset{\_}{S}} \rbrack}\Delta \quad {\overset{\_}{\hat{\sigma}}( {\rho_{n}^{*},z_{m}^{*}} )}}}}$

where N′≦N to ensure this system of equations is not underdetermined.The unknown values Δ{circumflex over ({overscore (σ)})}(ρ_(n) ^(*),z_(m) ^(*)) can then be determined by solving the above set of linearequations. It is apparent that the embodiment described in the sectionentitled “REALIZATION OF THE TRANSFORMATION” is a special case of theabove for which M=0, N=N′=1.

Although the approximation Δ{tilde over (σ)}(ρ,z)=Δ{circumflex over(σ)}(ρ_(n) ^(*),z_(m) ^(*)) (which merely states that Δ{tilde over(σ)}(ρ,z) is a piecewise constant function of ρ,z) is used in theimmediately above embodiment, such an approximation is not necessary.More generally, it is possible to expand Δ{tilde over (σ)}(ρ,z) using aset of basis functions, and to then solve the ensuing set of equationsfor the coefficients of the expansion. Specifically, suppose${\Delta \quad {\overset{\sim}{\sigma}( {\rho,z} )}} = {\sum\limits_{m = {- \infty}}^{\infty}{\sum\limits_{n = {- \infty}}^{\infty}{a_{mn}{\varphi_{mn}( {\rho,z} )}}}}$

then,${\overset{\_}{v}}_{k} = {\sum\limits_{m = {- \infty}}^{\infty}{\sum\limits_{n = {- \infty}}^{\infty}{{I\lbrack {\overset{\_}{\overset{\_}{S}}\quad \varphi_{mn}} \rbrack}{\overset{\_}{a}}_{mn}}}}$

where {overscore (a)}_(mn)=a_(mn)[1,1, . . . ,1]^(T). Some desirableproperties for the basis functions φ_(mn), are: 1) the integralsI[{double overscore (S)}φ_(mn)] in the above equation all exist; and, 2)the system of equations for the coefficients a_(mn) is not singular. Itis helpful to select the basis functions so that a minimal number ofterms is needed to form an accurate approximation to Δ{tilde over(σ)}(ρ,z).

The above embodiment is a special case for which the basis functions areunit step functions. In fact, employing the expansion $\begin{matrix}{{\Delta \quad {\overset{\sim}{\sigma}( {\rho,z} )}} = \quad {\sum\limits_{m = {- M}}^{+ M}{\sum\limits_{n = 1}^{N^{\prime}}{\Delta \quad {{\overset{\_}{\hat{\sigma}}( {\rho_{n}^{*},z_{m}^{*}} )}\lbrack {{u( {z - z_{m}} )} -} }}}}} \\{ \quad {u( {z - z_{m - 1}} )} \rbrack \lbrack {{u( {\rho - \rho_{n}} )} - {u( {\rho - \rho_{n - 1}} )}} \rbrack}\end{matrix}$

where u(.) denotes the unit step function leads directly to the samesystem of equations${\overset{\_}{v}}_{k} = {\sum\limits_{m = {- M}}^{+ M}{\sum\limits_{n = 1}^{N^{\prime}}{{I_{mn}\lbrack \overset{\_}{\overset{\_}{S}} \rbrack}\Delta \quad {\overset{\_}{\hat{\sigma}}( {\rho_{n}^{*},z_{m}^{*}} )}}}}$

given in the above embodiment. Specific values for M,N′, z_(m), andρ_(n) needed to realize this embodiment of the invention depend on theexcitation frequency(ies), on the transmitter-receiver spacings that areunder consideration, and generally on the background conductivity anddielectric constant. Different values for z_(m) and ρ_(n) are generallyused for different depth intervals within the same well because thebackground medium parameters vary as a function of depth in the well.

Solving the immediately above system of equations results in estimatesof the average conductivity and dielectric constant within the volume ofthe earth formation 215 corresponding to each integral I_(mn)[{doubleoverscore (S)}]. In an embodiment, the Least Mean Square method is usedto determine values for Δ{circumflex over ({overscore (σ)})}(ρ_(n)^(*),z_(m) ^(*)) by solving the above system of equations. Many texts onlinear algebra list other techniques that may also be used.

Unlike other procedures previously used for processing MWD/LWD data, thetechniques of a disclosed embodiment account for dielectric effects andprovide for radial inhomogeneities in addition to bedding interfaces byconsistently treating the signal as a complex-valued function of theconductivity and the dielectric constant. This procedure producesestimates of one variable (i.e., the conductivity) are not corrupted byeffects of the other (i.e., the dielectric constant). As mentioned inthe above “SUMMARY OF THE INVENTION,” this result was deemedimpracticable as a consequence of the “old assumptions.”

A series of steps, similar to those of FIG. 9, can be employed in orderto implement the embodiment for Multiple Sensors at Multiple Depths.Since the lookup table for I_(mn)[{double overscore (S)}] needed torealize such an embodiment could be extremely large, these values areevaluated as needed in this embodiment. This can be done in a manneranalogous to the means described in the above section “REALIZATION OFTHE TRANSFORMATION” using the following formulae:${{{I_{mn}\lbrack S\rbrack} = {\frac{1}{w_{0}}\frac{\partial w}{\partial{\overset{\sim}{\sigma}}_{mn}}}}}_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}}$

${\frac{\partial w}{\partial{\overset{\sim}{\sigma}}_{mn}}}_{\overset{\sim}{\sigma} = {\overset{\sim}{\sigma}}_{0}} = {\lim\limits_{{\Delta \quad {\overset{\sim}{\sigma}}_{mn}}arrow 0}{\frac{{w( {{\overset{\sim}{\sigma}}_{0} + {\Delta \quad {\overset{\sim}{\sigma}}_{mn}}} )} - {w( {\overset{\sim}{\sigma}}_{0} )}}{( {{\overset{\sim}{\sigma}}_{0} + {\Delta \quad {\overset{\sim}{\sigma}}_{mn}}} ) - {\overset{\sim}{\sigma}}_{0}}.}}$

where {tilde over (σ)}_(mn)=σ_(mn)+iωε_(mn) represents the conductivityand dielectric constant of the region of space over which the integralI_(mn)[S] is evaluated. In words, I_(mn)[S] can be calculated byevaluating the derivative of the measurement with respect to the mediumparameters within the volume covered by the integration. Alternatively,one could evaluate I_(mn)[S] by directly carrying out the integration asneeded. This eliminates the need to store the values in a lookup table.

While the above exemplary systems are described in the context of anMWD/LWD system, it shall be understood that a system according to thedescribed techniques can be implemented in a variety of other loggingsystems such as wireline induction or wireline dielectric measurementsystems. Further in accordance with the disclosed techniques, it shouldbe understood that phase shift and attenuation can be combined in avariety of ways to produce a component sensitive to resistivity andrelatively insensitive to dielectric constant and a component sensitiveto dielectric constant and relatively insensitive to resistivity. In theinstance of MWD/LWD resistivity measurement systems, resistivity is thevariable of primary interest; as a result, phase shift and attenuationmeasurements can be combined to produce a component sensitive toresistivity and relatively insensitive dielectric constant.

Single Measurements at a Single Depth

One useful embodiment is to correlate (or alternatively equate) a singlemeasured value w₁ to a model that predicts the value of the measurementas a function of the conductivity and dielectric constant within aprescribed region of the earth formation. The value for the dielectricconstant and conductivity that provides an acceptable correlation (oralternatively solves the equation) is then used as the final result(i.e., correlated to the parameters of the earth formation). Thisprocedure can be performed mathematically, or graphically. Plotting apoint on a chart such as FIG. 8 and then determining which dielectricvalue and conductivity correspond to it is an example of performing theprocedure graphically. It can be concluded from the preceding sections,that Ŝ is the sensitivity of such an estimate of the dielectric constantand conductivity to perturbations in either variable. Thus such aprocedure results in an estimate for the conductivity that has no netsensitivity to the changes in the dielectric constant and an estimatefor the dielectric constant that has no net sensitivity to changes inthe conductivity within the volume in question. This is a very desirableproperty for the results to have. The utility of employing a singlemeasurement at a single depth derives from the fact that data processingalgorithms using minimal data as inputs tend to provide results quicklyand reliably. This procedure is a novel means of determining oneparameter (either the conductivity or the dielectric constant) with nonet sensitivity to the other parameter. Under the old assumptions, thisprocedure would appear to not be useful for determining independentparameter estimates.

Iterative Forward Modeling and Dipping Beds

The analysis presented above has been carried out assuming a2-dimensional geometry where the volume P 225 in FIG. 2 is a solid ofrevolution about the axis of the tool. In MWD/LWD and wirelineoperations, there are many applications where such a 2-dimensionalgeometry is inappropriate. For example, the axis of the tool oftenintersects boundaries between different geological strata at an obliqueangle. Practitioners refer to the angle between the tool axis and avector normal to the strata as the relative dip angle. When the relativedip angle is not zero, the problem is no longer 2-dimensional. However,the conclusion that: 1) the attenuation measurement is sensitive to theconductivity in the same volume as the phase measurement is sensitive tothe dielectric constant; and, 2) an attenuation measurement is sensitiveto the dielectric constant in the same volume that the phase measurementis sensitive to the conductivity remains true in the more complicatedgeometry. Mathematically, this conclusion follows from theCauchy-Reimann equations which still apply in the more complicatedgeometry (see the section entitled “SENSITIVITY FUNCTIONS”). Thephysical basis for this conclusion is that the conduction currents arein quadrature (90 degrees out of phase) with the displacement currents.At any point in the formation, the conduction currents are proportionalto the conductivity and the displacement currents are proportional tothe dielectric constant.

A common technique for interpreting MWD/LWD and wireline data inenvironments with complicated geometry such as dipping beds is to employa model which computes estimates for the measurements as a function ofthe parameters of a hypothetical earth formation. Once model inputparameters have been selected that result in a reasonable correlationbetween the measured data and the model data over a given depthinterval, the model input parameters are then correlated to the actualformation parameters. This process is often referred to as “iterativeforward modeling” or as “Curve Matching,” and applying it in conjunctionwith the old assumptions, leads to errors because the volumes in whicheach measurement senses each variable have to be known in order toadjust the model parameters appropriately.

The algorithms discussed in the previous sections can also be adaptedfor application to data acquired at non-zero relative dip angles.Selecting the background medium to be a sequence of layers having theappropriate relative dip angle is one method for so doing.

Transformations for a Resitivity-Dependent Dielectric Constant

In the embodiments described above, both the dielectric constant andconductivity are treated as independent quantities and the intent is toestimate one parameter with minimal sensitivity to the other. As shownin FIG. 1, there is empirical evidence that the dielectric constant andthe conductivity can be correlated. Such empirical relationships arewidely used in MWD/LWD applications, and when they hold, one parametercan be estimated if the other parameter is known.

This patent application shows that: 1) an attenuation measurement issensitive to the conductivity in the same volume of an earth formationas the phase measurement is sensitive to the dielectric constant; and,2) the attenuation measurement is sensitive to the dielectric constantin the same volume that the phase measurement is sensitive to theconductivity. A consequence of these relationships is that it is notgenerally possible to derive independent estimates of the conductivityfrom a phase and an attenuation measurement even if the dielectricconstant is assumed to vary in a prescribed manner as a function of theconductivity. The phrase “not generally possible” is used above becauseindependent estimates from each measurement can be still be made if thedielectric constant doesn't depend on the conductivity or if theconductivity and dielectric constant of earth formation are practicallythe same at all points within the sensitive volumes of bothmeasurements. Such conditions represent special cases which are notrepresentative of conditions typically observed within earth formations.

Even though two independent estimates of the conductivity are notgenerally possible from a single phase and a single attenuationmeasurement, it is still possible to derive two estimates of theconductivity from a phase and an attenuation measurement given atransformation to convert the dielectric constant into a variable thatdepends on the resistivity. For simplicity, consider a device such asthat of FIG. 3. Let the complex number w, denote an actual measurement(i.e., the ratio of the voltage at receiver 307 relative to the voltageat receiver 309, both voltages induced by current flowing throughtransmitter 305). Let the complex number w denote the value of saidmeasurement predicted by a model of the tool 320 in a prescribed earthformation 315. For further simplicity, suppose the model is as describedabove in the section “REALIZATION OF THE TRANSFORMATION.” Then,${w \equiv {w( {\sigma,{ɛ(\sigma)}} )}} = {( \frac{L_{1}}{L_{2}} )^{3}\frac{{\exp ( {ikL}_{2} )}( {1 - {ikL}_{2}} )}{{\exp ( {ikL}_{1} )}( {1 - {ikL}_{1}} )}}$

where the wave number k≡k(σ,ε(σ))={square root over (iωμ(σ+iωε(σ)))},and the dependence of the dielectric constant ε on the conductivity σ isaccounted for by the function ε(σ). Different functions ε(σ) can beselected for different types of rock. Let σ_(P) and σ_(A) denote twoestimates of the conductivity based on a phase and an attenuationmeasurement and a model such as the above model. The estimates can bedetermined by solving the system of equations

0=|w ₁ |−|w(σ_(A),ε(σ_(P)))|

0=phase(w ₁)−phase(w(σ _(P),ε(σ_(A)))).

The first equation involves the magnitude (a.k.a. the attenuation) ofthe measurement and the second equation involves the phase (a.k.a. thephase shift) of the measurement. Note that the dielectric constant ofone equation is evaluated using the conductivity of the other equation.This disclosed technique does not make use of the “old assumptions.”Instead, the attenuation conductivity is evaluated using a dielectricvalue consistent with the phase conductivity and the phase conductivityis evaluated using a dielectric constant consistent with the attenuationconductivity. These conductivity estimates are not independent becausethe equations immediately above are coupled (i.e., both variables appearin both equations). The above described techniques represent asubstantial improvement in estimating two resistivity values from aphase and an attenuation measurement given a priori information aboutthe dependence of the dielectric constant on the conductivity. It can beshown that the sensitivity functions for the conductivity estimatesσ_(A) and σ_(P) are S′/I[S′] and S″/I[S″], respectively if theperturbation to the volume P 225 is consistent with the assumeddependence of the dielectric constant on the conductivity andσ_(A)=σ_(P).

It will be evident to those skilled in the art that a more complicatedmodel can be used in place of the simplifying assumptions. Such a modelmay include finite antennas, metal or insulating mandrels, formationinhomogeneities and the like. In addition, other systems of equationscould be defined such as ones involving the real and imaginary parts ofthe measurements and model values. As in previous sections of thisdisclosure, calibration factors and borehole corrections may be appliedto the raw data.

Transformations for a Resistivity-Dependent Dielectric Constant used inConjunction with Conventional Phase Resistivity Values

The preceding section, “TRANSFORMATIONS FOR A RESISTIVITY-DEPENDENTDIELECTRIC CONSTANT,” provides equations for σ_(P) and σ_(A) that do notmake use of the “old assumptions.” One complication that results fromusing said equations is that both a phase shift measurement and anattenuation measurement must be available in order to evaluate eitherσ_(P) or σ_(A). In MWD/LWD applications, both phase shift andattenuation measurements are commonly recorded; however, the attenuationmeasurements are often not telemetered to the surface while drilling.Since bandwidth in the telemetry system associated with the logging toolis limited, situations arise where it is useful to have less preciseresistivity measurements in favor of other data such as density, speedof sound or directional data. The following reparameterization of thesystem of equations in the previous section accommodates this additionalconsideration:

0=|w ₁ |−|w(σ_(A),ε(σ_(P)))|

0=phase(w ₁)−phase(w(σ_(P),ε(σ_(A)))),

where σ_(P) and σ_(A) respectively represent the reciprocals of thefirst and second resistivity values; the second equation involves aphase of the measured electrical signal and the first equation involvesa magnitude of the measured electrical signal at a given frequency ofexcitation; a function ε(.) represents a correlation between thedielectric constant and the resistivity and is evaluated in σ_(P) in thefirst and second equations; w₁ represents an actual measurement (e.g.,the ratio of the voltage at receiver 307 relative to the voltage atreceiver 309, both voltages induced by current flowing throughtransmitter 305) in the form of a complex number, and a function w(σ,ε)represents a mathematical model which estimates w₁. More particularly,ε(σ_(P)) represents the transformation of dielectric constant into avariable that depends upon resistivity.

The second equation evaluates phase conductivity σ_(P) and thedielectric constant correlation with the same phase conductivity whichis consistent with an assumption that the phase measurement senses bothresistivity and dielectric constant in substantially the same volume.The first equation evaluates attenuation conductivity σ_(A) and thedielectric constant correlation with the phase conductivity (not anattenuation conductivity) which is consistent with the attenuationmeasurement sensing the resistivity and dielectric constant in differentvolumes. The second equation allows σ_(P) to be determined by a phaseshift measurement alone. The first equation is actually the sameequation as in the previous section, but the value of σ_(A) that solvesthe equation may not be the same as in the previous section because thevalue of σ_(P) used in the correlation between the dielectric constantand the resistivity may not be the same. In other words, the phaseconductivity σ_(P) can be determined without the attenuationmeasurement, but the attenuation conductivity σ_(A) is a function ofboth attenuation and phase shift measurements. The value of σ_(P)resulting from these equations is the conventional phase conductivitycited in the prior art which is derived using the “old assumptions” thatthe phase shift measurement senses both the resistivity and dielectricconstant within substantially the same volume. The value for σ_(A) isconsistent with the fact the attenuation measurement senses theresistivity in substantially the same volume that the phase measurementsenses the dielectric constant; however, the σ_(A) value is slightlyless accurate than the σ_(A) values in the previous section because thephase conductivity σ_(P) used to evaluate the correlation between thedielectric constant and the resistivity is derived using the “oldassumptions.” The above two equations are partially coupled in that thephase conductivity σ_(P) is needed from the second equation to determinethe attenuation conductivity σ_(A) in the first equation.

Since the phase shift measurement is typically less sensitive to thedielectric constant than the corresponding attenuation measurement, theequations given in this section for σ_(P) and σ_(A) provide a reasonabletradeoff between the following: 1) accurate resistivity measurementresults, 2) consistent resistivity measurement results (reporting thesame resistivity values from both recorded and telemetered data), and 3)minimizing amounts of telemetered data while drilling.

When the perturbation to the volume P 225 is consistent with the assumeddependence of the dielectric constant on the conductivity, thesensitivity functions for conductivity estimates σ_(P) and σ_(A)determined by the equations given in this section are:${\Delta\sigma}_{P} = {\frac{S^{''} - {\omega \frac{ɛ}{\sigma}S^{\prime}}}{{I\lbrack S^{''} \rbrack} - {\omega \quad \frac{ɛ}{\sigma}{I\lbrack S^{\prime} \rbrack}}}{\Delta\sigma\Delta\rho\Delta}\quad z}$${\Delta\sigma}_{A} = {\lbrack {\frac{S^{\prime}}{I\lbrack S^{\prime} \rbrack} + \frac{( {\omega \frac{ɛ}{\sigma}} )^{2}( {{S^{\prime}{I\lbrack S^{''} \rbrack}} - {S^{''}{I\lbrack S^{\prime} \rbrack}}} )}{{I\lbrack S^{\prime} \rbrack}( {{I\lbrack S^{''} \rbrack} - {\omega \frac{ɛ}{\sigma}{I\lbrack S^{\prime} \rbrack}}} )}} \rbrack {\Delta\sigma\Delta}\quad \rho \quad \Delta \quad z}$

where $\frac{ɛ}{\sigma}$

represents the derivative of function ε(σ), and it was assumed forsimplicity that σ_(A)=σ_(P). The sensitivity function for σ_(P) is thecoefficient of ΔσΔρΔz in the first equation, and the sensitivityfunction for σ_(A) is the coefficient of ΔσΔρΔz in the second equation.Note that the leading term$\frac{S^{''}}{I\lbrack S^{''} \rbrack}$

in the sensitivity function of the first equation and the leading term$\frac{S^{\prime}}{I\lbrack S^{\prime} \rbrack}$

in the sensitivity function of the second equation are approximately thesensitivity function mentioned in the section, “TRANSFORMATIONS FOR ARESISTIVITY-DEPENDENT DIELECTRIC CONSTANT,” and that the remaining termsare representative of the sensitivity error resulting from using theequations given in this section. In the second equation, the error termin the sensitivity for σ_(A) tends to be small because it isproportional to $( {\omega \frac{ɛ}{\sigma}} )^{2}.$

It can be shown that the error term for the sensitivity function of aσ_(A) estimate resulting from equations consistent with the oldassumptions is proportional to$( {\omega \frac{ɛ}{\sigma}} )$

and is thus a larger error than produced by the technique described inthis section. As a result, the technique of this section represents asubstantial improvement over the prior art.

Approximations to Facilitate use of Rapidly-Evaluated Models

Many of the techniques described in the previous sections are even moreuseful if they can be applied in conjunction with a simplified model ofthe measurement device. This is especially true for embodiments whichuse iterative numerical techniques to solve systems of nonlinearequations because the amount of computer time required to achieve asolution is reduced if details of the measurement device can be ignored.Such details include finite sized antennas 205, 207 and a metallic drillcollar 203. Models that do not include such details can often beevaluated rapidly in terms of algebraic functions whereas modelsincluding these details may require a numerical integration or othersimilar numerical operations. An alternative choice for solving suchsystems of equations is to store multidimensional lookup tables andperform inverse interpolation. An advantage of using the lookup tablesis that the model calculations are done in advance; so, results can bedetermined quickly once the table has been generated. However, 1) alarge amount of memory may be required to store the lookup table; 2) itmay be difficult to handle inputs that are outside the range of thetable; 3) the tables may require regeneration if the equipment ismodified, and 4) the tables themselves may be costly to generate andmaintain.

The approach taken here is a compromise. A one-dimensional lookup tableis used to renormalize each measurement so as to be approximatelyconsistent with data from a simplified measurement device which hasinfinitesimal antennas and no metallic mandrel. The renormalized dataare used in conjunction with a simpler, but more rapidly evaluated modelto determine the conductivity and/or dielectric constant estimates.Higher dimensional lookup tables could be used to account for morevariables, but this has proved unnecessary in practice.

Suppose the function h₀ (σ,ε) represents a model that estimates themeasurements as a function of the conductivity and dielectric constantwhich includes details of the tool that are to be normalized away (i.e.finite antennas and a metallic mandrel). Suppose h₁(σ,ε) is a simplifiedmodel which is a function of the same formation parameters (i.e. theresistivity and the dielectric constant) but that assumes infinitesimalantennas and no metallic mandrel. The data shown in FIG. 10 are for themedium spaced 2 MHz measurement described in conjunction with FIG. 7.The first two columns of data resistivity (1/sigma) and dielectricconstant (eps_rel) values are input into the respective models (whichare not used in the algorithm, but are shown to illustrate how the tableis generated). It was found that satisfactory results can be achieved bycalculating the data as a function of the conductivity for only onedielectric constant, and in this embodiment, a relative dielectricconstant of 30 was used. The third column (db_pt) contains theattenuation values for the simplified model evaluated as a function ofconductivity, and the fourth column (db_man) contains the attenuationvalues for the model which explicitly accounts for the finite sizedantennas and metallic mandrel. The fifth (deg_pt) and sixth (deg_man)columns are similar to the third and fourth columns but for the phaseshift instead of the attenuation. All data are calibrated to read zeroif the electrical parameters of the surrounding medium are substantiallythat of a vacuum (i.e., the air) (σ=0,ε=1). Specifically, columns 3 and4 are:

db _(—) pt=20log ₁₀(h ₁(σ,ε)/h ₁(0,1))

db _(—) man=20log ₁₀(h ₀(σ,ε)/h ₀(0,1))

Columns 5 and 6 are:${\deg \quad \_ \quad p\quad t} = {\frac{180}{\pi}{\arg ( {{h_{1}( {\sigma,ɛ} )}/{h_{1}( {0,1} )}} )}}$${\deg \quad \_ \quad {man}} = {\frac{180}{\pi}{{\arg ( {{h_{0}( {\sigma,ɛ} )}/{h_{0}( {0,1} )}} )}.}}$

In practice, calibrated measurements are used as values for db_man orphs_man (i.e. the ordinates in the one-dimensional interpolation). Thecorresponding values for db_pt or phs_pt result from the interpolation.Once the values for db_pt or phs_pt have been determined, relativelysimple models such as$h_{1} = {( \frac{L_{1}}{L_{2}} )^{3}\quad \frac{{\exp ( {{ik}_{0}L_{2}} )}( {1 - {{ik}_{0}L_{2}}} )}{{\exp ( {{ik}_{0}L_{1}} )}( {1 - {{ik}_{0}L_{1}}} )}}$

can be parameterized as described in the several preceding sections andused to solve numerically for the desired parameters. The solutions canbe obtained quickly, using robust and well known numerical means such asthe nonlinear least squares technique.

An alternative procedure would be to transform the values from therelatively simple model to corresponding db_man and deg_man values. Thiswould produce equivalent results, but in conjunction with an iterativesolution method, is clumsy because it requires the values from thesimple model to be converted to db_man and deg_man values at each stepin the iteration.

The foregoing disclosure and description of the various embodiments areillustrative and explanatory thereof, and various changes in thedescriptions, modeling and attributes of the system, the organization ofthe measurements, transmitter and receiver configurations, and the orderand timing of steps taken, as well as in the details of the illustratedsystem may be made without departing from the spirit of the invention.

I claim:
 1. A method of determining a first electrical parameter of anearth formation through which a borehole is drilled, the methodcomprising the steps of: transforming a second electrical parameter ofthe earth formation into a variable that depends on the first electricalparameter; estimating a first value for the first electrical parameterfrom the properties of a measured electrical signal consistent with thetransforming step and consistent with an assumption that each propertyof the measured electrical signal senses the first electrical parameterand the second electrical parameter in substantially the same volume;and, estimating a second value for the first electrical parameter fromthe properties of the measured electrical signal consistent with thetransforming step, consistent with the estimated first value for thefirst electrical parameter, and also consistent with each property ofthe measured electrical signal sensing the first electrical parameterand the second electrical parameter in different volumes.
 2. The methodof claim 1, wherein the measured electrical signal comprises anattenuation measurement and a phase shift measurement between a firstreceiver coil and a second receiver coil.
 3. The method of claim 1,wherein the first electrical parameter comprises a resistivity of theearth formation and the second electrical parameter comprises adielectric constant of the earth formation.
 4. The method of claim 3,wherein the first value is determined by solving a first equation,0=phase(w₁)−phase(w(σ_(P),ε(σ_(p)))), and the second value is determinedby solving a second equation, 0=|w₁|−|w(σ_(A),ε(σ_(p)))|, where σ_(P)and σ_(A) respectively represent the reciprocals of the first and secondvalues; the first equation involves a phase shift measurement of themeasured electrical signal and the second equation involves anattenuation measurement of the measured electrical signal at a givenfrequency of excitation; a function ε(.) represents a correlationbetween the dielectric constant and the resistivity and is evaluated inσ_(P) in the first and second equations; w₁ represents an actualmeasurement in the form of a complex number, and a function w(σ,ε)represents a model which estimates w₁.
 5. The method of claim 1, whereinthe properties of the measured signal used to estimate the first valueof the first electrical parameter are less sensitive to the secondelectrical parameter than the properties of the measured signal used toestimate the second value of the first electrical parameter.
 6. Themethod of claim 1 further comprising the steps of: transforming themeasured electrical signal to be consistent with a hypotheticalelectrical signal that would be measured by a hypothetical measurementdevice measurement simpler than an actual measurement device; and,employing a model of the hypothetical measurement device to perform theestimating steps.
 7. The method of claim 6, wherein the step oftransforming the measured electrical signal to be consistent with asignal that would be measured by a hypothetical measurement devicecomprises the steps of: generating a lookup table containing a first setof values representative of the measured electrical signal and a secondset of values representative of the hypothetical electrical signal as afunction of at least one electrical parameter; and, using the lookuptable to interpolate estimates for the hypothetical electrical signalgiven the measured electrical signal.
 8. The method of claim 6, whereinthe hypothetical measurement device is comprised of infinitesimalantennas and no mandrel.
 9. The method of claim 6, wherein the employingstep comprises the step of: normalizing either the measured electricalsignal or the hypothetical electrical signal to correspond to oneanother.
 10. A system for determining a first electrical parameter of anearth formation through which a borehole is drilled, comprising: a meansfor transforming a second electrical parameter of the earth formationinto a variable that depends on the first electrical parameter; a meansfor estimating a first value for the first electrical parameter from theproperties of a measured electrical signal consistent with thetransforming step and consistent with the assumption that each propertyof the measured electrical signal senses the first electrical parameterand the second electrical parameter in substantially the same volume;and a means for estimating a second value for the first electricalparameter from the properties of a measured electrical signal consistentwith the transforming step, consistent with the estimated first valuefor the first electrical parameter, and also consistent with eachproperty of the measured electrical signal sensing the first electricalparameter and the second electrical parameter in different volumes. 11.The system of claim 10, wherein the first electrical parameter comprisesa resistivity of the earth formation and the second electrical parametercomprises a dielectric constant of the earth formation.
 12. The systemof claim 11, wherein the first value is determined by solving a firstequation, 0=phase(w₁)−phase(w(σ_(P),ε(σ_(p)))), and the second value isdetermined by solving a second equation, 0=|w₁|−|w(σ_(A),ε(σ_(p)))|,where σ_(P) and σ_(A) respectively represent the reciprocals of thefirst and second values; the first equation involves a phase shiftmeasurement of the measured electrical signal and the second equationinvolves an attenuation measurement of the measured electrical signal ata given frequency of excitation; a function ε(.) represents acorrelation between the dielectric constant and the resistivity and isevaluated in σ_(P) in the first and second equations; w₁ represents anactual measurement in the form of a complex number, and a functionw(σ,ε) represents a model which estimates w₁.
 13. The system of claim10, wherein the measured electrical signal comprises an attenuationmeasurement and a phase shift measurement between a first receiver coiland a second receiver coil.
 14. The system of claim 10, wherein theproperties of the measured signal used to estimate the first value ofthe first electrical parameter are less sensitive to the secondelectrical parameter than the properties of the measured signal used toestimate the second value of the first electrical parameter.
 15. Thesystem of claim 14, wherein the properties of the measured signal usedto estimate the first value of the first electrical parameter comprise aphase shift measurement, the properties of the measured signal used toestimate the second value of the first electrical parameter comprise aphase shift measurement and an attenuation measurement.
 16. A method ofdetermining a first electrical parameter of an earth formation throughwhich a borehole is drilled, comprising the steps of: determining afirst value for the first electrical parameter based on a firstelectrical measurement and a correlation between the first electricalparameter and a second electrical parameter; and determining a secondvalue for the first electrical parameter based on a second electricalmeasurement, the first value for the first electrical parameter, and thecorrelation between the first electrical parameter and the secondelectrical parameter.
 17. The method of claim 16, wherein the firstelectrical parameter comprises a resistivity of the earth formation andthe second electrical parameter comprises a dielectric constant of theearth formation.
 18. The method of claim 16, wherein the firstelectrical measurement comprises a phase shift measurement and thesecond electrical measurement comprises an attenuation measurement. 19.The method of claim 16, wherein the first electrical measurement is lesssensitive to the second electrical parameter than the second electricalmeasurement.
 20. The method of claim 16, wherein the first value isdetermined based on an assumption that the first electrical measurementand the second electrical measurement sensing the first electricalparameter and the second electrical parameter in substantially the samevolume and the second value is determined based on the first electricalmeasurement and the second electrical measurement sensing the firstelectrical parameter and the second electrical parameter in differentvolumes.
 21. A computer-readable medium storing a software program, thesoftware program configured to enable a processor to perform the methodof claim
 1. 22. A computer-readable medium storing the first value andthe second value determined in accordance with the method of claim 1.23. A computer-readable medium storing a software program, the softwareprogram configured to enable a processor to perform the method of claim16.
 24. A computer-readable medium storing the first value and thesecond value determined in accordance with the method of claim 16.